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%I #58 Aug 31 2023 07:54:16
%S 0,1,3,19,189,2576,44683,941977,23388025,668520163,21622993111,
%T 780789908240,31135480907413,1358965445353621,64440211018897379,
%U 3298807094967155971,181322497435007616497,10651131815012588324380,665881649529214120845679,44144097851022253967955749
%N E.g.f. is series reversion of log(1+x)*exp(-x).
%C A simple grammar.
%C For n > 0, Sum_{k=1..n} a(k)*Sum_{i=0..n-k} (-1)^i*k^i*Stirling1(n-i,k)/(i!*(n-i)!) = delta(n,1). - _Vladimir Kruchinin_, Feb 08 2012
%C From _Gus Wiseman_, Jul 20 2013: (Start)
%C Number of tail-trees of weight n. A tail is a pairing of a block of a set partition p with an element of some other block. A tail-tree on p is composed of a root block r, a tail-tree on each block of a set partition of the remaining blocks, and a tail from each of their roots to r.
%C On any set partition of weight n and length m, the total number of tail-forests with k components is equal to binomial(m-1, k-1)*n^(m-k). (End)
%C From _Paul Laubie_, Aug 25 2023: (Start)
%C Number of nonempty forests of rooted labeled hypertrees with a total number of vertices equal to n.
%C E.g., n=1: Only one forest is possible, which is {1}, the forest with one hypertree with one vertex.
%C n=2: Three forests are possible: {1,2}, the forest with two hypertrees, each having one vertex labeled 1 for one on the hypertree and 2 for the other hypertree; the forest {1-2}, with only one hypertree, with two vertices rooted at 1; and the forest {2-1}, with only one hypertree, with two vertices rooted at 2. (End)
%H G. C. Greubel, <a href="/A052888/b052888.txt">Table of n, a(n) for n = 0..368</a>
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=863">Encyclopedia of Combinatorial Structures 863</a>
%H Rosena R. X. Du and Fu Liu, <a href="http://arxiv.org/abs/1008.3677">Pure-cycle Hurwitz factorizations and multi-noded rooted trees</a>, arXiv:1008.3677 [math.CO], 2010-2013.
%H Gus Wiseman, <a href="http://youtu.be/8SrkcERO0xA">All 189 tail-trees of weight 4.</a>
%H Gus Wiseman, <a href="http://dx.doi.org/10.1016/j.disc.2007.07.009">Set maps, umbral calculus, and the chromatic polynomial</a>, Discrete Math., 308(16):3551-3564, 2008.
%F E.g.f.: RootOf(_Z-exp(exp(_Z)*x)+1)
%F a(n) = Sum_{k=1..n} Stirling2(n, k)*n^(k-1). - _Vladeta Jovovic_, Jul 26 2005
%F a(n) = exp(-n)*Sum_{k>=1} n^(k-1)*k^n/k!. - _Vladeta Jovovic_, Jul 03 2006 [corrected by _Ilya Gutkovskiy_, Apr 20 2020]
%F a(n) ~ exp(n*(LambertW(1) + 1/LambertW(1) - 2)) * n^(n-1) / sqrt(1+LambertW(1)). - _Vaclav Kotesovec_, Jan 22 2014
%p spec := [S,{C=Prod(Z,B),B=Set(S),S=Set(C,1 <= card)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t Table[Sum[StirlingS2[n,k]*n^(k-1),{k,1,n}],{n,0,20}] (* _Vaclav Kotesovec_, Jan 22 2014 *)
%o (PARI) for(n=0,30, print1(sum(k=1,n, stirling(n,k,2)*n^(k-1)), ", ")) \\ _G. C. Greubel_, Nov 17 2017
%K easy,nonn
%O 0,3
%A encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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